Geodesic rigidity of conformal connections on surfaces

Abstract

Abstract. We show that a conformal connection on a closed oriented surface Σ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on Σ determine the metric up to constant rescaling. It is also shown that every conformal connection on the 2-sphere lies in a complex 5-manifold of conformal connections, all of which share the same unparametrised geodesics.

Extremal conformal structures on projective surfaces

“…Remark 5.5. In [38], it is shown that for a compact oriented projective surface (Σ, p) of negative Euler characteristic the functional E p admits at most one absolute minimiser [g] (i.e. a conformal structure [g] such that E p ([g]) = 0).…”

Projectively equivalent Finsler metrics on surfaces of negative Euler characteristic

“…The corresponding result for Riemannian metrics has been obtained in [7, Corollary 3] (see also [8,9,10]), where the assumption of real-analicity is not necessary.…”

Deformations of the Veronese Embedding and Finsler -Spheres of Constant Curvature

“…The Weyl connection has the property that all 4 C. Lange and T. Mettler of its maximal geodesics are embedded circles and hence define a Besse-Weyl structure. In addition, they show that every such Weyl connection on S 2 is part of a complex 5-dimensional family of Weyl connections having the same unparametrised geodesics (see also [32]). In particular, LeBrun and Mason's Weyl connections that arise from an embedding of RP 2 that is sufficiently close to the standard embedding provide examples of positive Besse-Weyl structures.…”